Outdated Archival Version

These pages are not updated anymore. They reflect the state of 20 August 2005. For the current production of this journal, please refer to http://www.math.psu.edu/era/.

\input amstex
\documentstyle{era-ams}
\document
%\magnification=1200
\define\ZZ{{\Bbb Z}}
\define\TT{{\Bbb T}}
\define\CC{{\Bbb C}}
\define\RR{{\Bbb R}}
\define\NN{{\Bbb N}}
\define\QQ{{\Bbb Q}}
\define\bv{{\bold v}}
\define\bu{{\bold u}}
\define\bw{{\bold w}}
\define\bR{{\bold R}}
\define\Lapl{{\bold \Delta}}
\define\cR{{\Cal R}}
\define\bom{{\bold \omega}}
\define\bal{{\bold \alpha}}
\define\bzero{{\bold 0}}
\define\btau{{\bold \tau}}
\define\dvol{d\,\hbox{vol}}

\topmatter
\title
On Quantum Limits on Flat Tori
\endtitle
\author Dmitry Jakobson
\endauthor
\address
Department of Mathematics,
Princeton University,
Princeton, NJ 08544
\endaddress
\email  diy\@math.princeton.edu \endemail
\date April 20, 1995\inRevisedForm July 19, 1995 \enddate
\communicatedby{Yitzhak Katznelson}
\abstract
We classify all weak $*$ limits of squares of normalized eigenfunctions of 
the Laplacian on two-dimensional flat tori (we call these limits
{\it quantum limits}).  We also obtain several results about such limits in
dimensions three and higher.  Many of the results are a consequence of a
geometric lemma which describes a property of simplices of codimension
one in $\RR^n$ whose vertices are lattice points on spheres.
The lemma follows from the finiteness of the number of solutions
of a system of two Pell equations.  A consequence of the lemma is a
generalization of the result of B. Connes.  We also indicate a proof
(communicated to us by J. Bourgain) of the absolute continuity of the
quantum limits on a flat torus in any dimension.   We generalize
a two-dimensional result of Zygmund to three dimensions; we discuss various
possible generalizations of that result to higher dimensions and the
relation to $L^p$ norms of the densities of quantum limits and
their Fourier series.
\endabstract
\cvol{1}
\cvolno{2}
\cvolyear{1995}
\cyear{1995}
%\keywords
%flat tori, Laplacian, resonances, Pell equation, weak $*$ limits, 
%Fourier series
%\endkeywords
%\subjclass
%42B05, 81Q50, 58C40, 52B20, 11D09, 11J86
%\endsubjclass

\footnote""{\noindent 1991 {\it Mathematics Subject Classification:}
42B05, 81Q50, 58C40, 52B20, 11D09, 11J86}
\footnote""{\noindent {\it Key words and phrases:} flat tori, Laplacian,
resonances, Pell equation, weak $*$ limits, Fourier series}
\endtopmatter
\document

\advance\pageno by 79

In this note we shall consider eigenfunctions of the Laplacian
on the standard flat $d$-dimensional torus $\TT^d = \RR^d/ \ZZ^d$.  
Most of the results of this paper are also valid for other flat tori 
$\RR^d/ \Lambda$, where $\Lambda$ is a lattice (of rank $d$) in $\RR^d$ 
different from $\ZZ^d$.  

Let $\varphi$ be an eigenfunction with the eigenvalue $\lambda$, and let 
$||\varphi||_2 = 1$.  The multiplicity of $\lambda$ is equal to the number of 
lattice points on a sphere of radius $\sqrt{\lambda}$ centered at the origin 
(or, equivalently, 
to the number of ways of writing $\lambda$ as a sum of $d$ squares).  
It is well-known that the multiplicity is unbounded for $d\geq 2$.

We are interested in all possible weak limits as $\lambda_j\to\infty$ of
the (probability) measures
$d\mu_j=|\varphi_j|^2 \dvol$ (where $\dvol$ is the Riemannian volume form
on $\TT^d$), which are henceforth called quantum limits (cf. [Jak]).

A. Zygmund proved in [Zyg] that for $d=2$
$$
||\varphi ||_4/||\varphi ||_2\ \leq\ 5^{1/4} \tag 1
$$
It follows that for $d=2$ any quantum limit $d\nu$ is absolutely continuous,
and its density is a function $\psi\in L^2$ satisfying
$||\psi ||_2\leq 5^{1/2}$.  In fact, much more can be said about the densities
of quantum limits on two-dimensional tori.

To formulate our result, denote the Fourier expansion of a quantum limit
$d\nu$ on $\TT^d$ by
$$
d\nu\ \sim\ \sum_{\btau\in \ZZ^d} c_\btau e^{i\; (\btau ,x)}, \tag 2
$$
where $(\btau ,x)$ is the usual scalar product in $\RR^d$.

We prove the following result:
\proclaim{Theorem 1}
The density of every quantum limit $d\nu$ on $\TT^{2}$
is a trigonometric polynomial all of whose non-zero frequencies lie on the 
union of (at most) two  circles centered at the origin.
\endproclaim
In other words, for every $d\nu$ with the Fourier expansion (2) there 
exist two positive numbers $r_1$ and $r_2$ such if  $\btau\neq\bzero$ and
$c_\btau\neq 0$ then either $|\btau |=r_1$ or $|\btau |=r_2$.  
We can also describe the combinations of $\btau$-s (as above) that occur
(together with the set of possible $c_\btau$-s).

The first question one asks for $d>2$ is whether quantum limits are absolutely
continuous with respect to Lebesgue measure or not.  No analogue of (1) is
known for $d>2$, so the absolute continuity of quantum limits has to be proved
differently than for $d=2$.
J. Bourgain has communicated to us the proof of
\proclaim{Theorem 2 (Bourgain)}
Quantum limits are absolutely continuous in any dimension.
\endproclaim

The proof involves repeated applications of
a result due to B. Connes\footnote{J. Bourgain has remarked that
Proposition 3 also follows from lemma 25 in [Bou2].} (see Proposition 3 below);
a property\footnote{Sometimes referred to as a "sequence splitting lemma".}
of sequences of nonnegative numbers uniformly bounded in $l^1$ is also used.

The rest of this note is concerned with
obtaining more precise information about the possible quantum limits in
dimensions three and higher, although we are not able to classify all such
limits as in dimension two.

The first result is
\proclaim{Theorem 3}
For $3\leq d\leq 5$, every quantum limit on $\TT^d$ satisfies
$$
\sum_{\btau\in\ZZ^d} |c_\btau|^{d-2}\ < \ \infty  \tag 3
$$
(where $c_\btau$-s are defined by (2)).
\endproclaim
We conjecture that (3) holds in any dimension $d>5$, but are not able to
prove it because of certain technical difficulties.  

It follows from Theorem 3 that the density of any
quantum limit on $\TT^3$ is a function that has absolutely convergent
Fourier series; for $d=4$ it follows that any such function is in $L^2$.
We also deduce from a certain conjecture about the distribution of $n$-tuples
of primes\footnote{A weak form of Hardy-Littlewood's conjecture, cf. [HL].}
(which we shall call {\it conjecture HL}, cf. [Jak]) that on a
three-dimensional torus there exist quantum limits which are not
trigonometric polynomials; on a four-dimensional torus we can prove this
unconditionally.

By Theorem 2, every quantum limit has a density which is a (nonnegative)
function in $L^1(\TT^d)$.
The question arises whether the densities of quantum limits on $\TT^d$
belong to $L^p$ for any $p>1$ and $d\geq 5$.  The answer turns out to be
intimately
related to the generalization of Zygmund's result (1) to higher dimensions.

Namely, we define the number $A(d,p)\leq\infty$ by
$$
A(d,p)\ =\ \sup_{\Lapl \varphi+\lambda\varphi=0}
( ||\varphi||_{p} / ||\varphi||_{2} ) \tag 4
$$
(where the supremum is taken over the eigenfunctions of the Laplacian
on $\TT^d$).\footnote{
The question of the rate of growth of the ratio in (4) with increasing
eigenvalue is one of the basic questions in quantum chaos, cf. [Sar].}
It is not known whether $A(d,p)<\infty$ for any $p>2$ and $d\geq 3$.
Bourgain showed in [Bou1] that $A(d,p)=\infty$ for
$d\geq 4$ and $p\geq 2(d+1)/(d-3)$.

We can show that if $A(d,p)=\infty$ for some  $p>2$, then there exist
quantum limits on $\TT^{d+3}$ whose densities are not in $L^{p/2}(\TT^{d+3})$.

To state the converse result, for $d\geq 2$ we define $B(d,p)$ by
$$
B(d,p)\ =\ \sup\left\{ ||\varphi||_{p} / ||\varphi||_{2}\ :\
\widehat{\varphi}\in(\Omega\cap\ZZ^{d+2}) \right\}
$$
where the supremum is taken over the trigonometric polynomials on $\TT^{d+2}$
all of whose frequencies lie on an intersection of $\ZZ^{d+2}$ with a
$(d-1)$-dimensional sphere $\Omega$ lying in a $d$-dimensional subspace of
$\RR^{d+2}$, and the $L^p$ norms are for the functions on $\TT^{d+2}$.
It is easy to see\footnote{We can identify the eigenfunctions on $\TT^d$ 
with the eigenfunctions on $\TT^{d+2}$ all of whose frequencies lie in the 
subspace $\{x\in\RR^{d+2}|x_{d+1}=x_{d+2}=0\}$ of $\RR^{d+2}$.} that 
$B(d,p)\geq A(d,p)$.  One can generalize (1) and prove that $B(2,4)<\infty$.  
It may happen that $B(d,p)<\infty$ whenever $A(d,p)<\infty$, 
but we cannot prove this.  

We can show that if $B(d,p)<\infty$ for some $d$ and $p>2$, then the
density of any quantum limit on $\TT^{d+2}$ is in $L^{p/2}(\TT^{d+2})$.

Let $\varphi$ be the eigenfunction of the Laplacian with the eigenvalue 
$\lambda$.  We want to prove an analogue of (1) for the Fourier series of the 
function $|\varphi|^2$.  The Fourier expansion of $\varphi$ is given by 
$$
\varphi(x)\ =\ \sum_{\xi\in \ZZ^d; |\xi |^2 = \lambda}
a_\xi e^{i\; (\xi ,x)},\ \ a_\xi\in\CC
$$
It follows from the previous formula that
$$
|\varphi|^2 = \sum_{\btau} b_\btau e^{i\; (\btau ,x)}, \ \
b_\btau =
\sum_{\scriptstyle\xi-\eta=\btau\atop
\scriptstyle |\xi |^2=|\eta |^2=\lambda}
a_\xi {\bar{a}}_\eta \tag 5
$$
If $A(d,p)<\infty$ for some $22$ because the exponent $q$ in our bound is bigger than two.
The ideas of the proof of Proposition 1 should allow to prove that
$C(d,d)<\infty$; however, for $d\geq 3$
certain technical difficulties arise which at the moment prevent us from
extending Proposition 1 to dimensions four and higher.

We can show that if $C(d,q)=\infty$, then there exist such quantum limits
on $\TT^{d+3}$ that the Fourier series of their densities are not in $l^q$.

To state the converse result, for $d\geq 2$ we define $D(d,q)$ by
$$
D(d,q)\ =\ \sup\left\{ ||\widehat{f}||_q / ||\varphi||_2^2\ :\
\widehat{\varphi}\in(\Omega\cap\ZZ^{d+2}) \right\} ,\ \ \ \ \ f=|\varphi|^2
$$
where the supremum is taken over the same trigonometric polynomials on
$\TT^{d+2}$ as in the definition of $B(d,p)$.
It is easy to see that $D(d,q)\geq C(d,q)$ and that   
for $2\leq p\leq 4$, $(B(d,p))^2\geq D(d,p/(p-2))$.  

We can show that if $D(d,q)<\infty$, then
for any quantum limit $d\nu$ on $\TT^{d+2}$ given by (2)
$$
\sum_{\btau\in\ZZ^{d+2}} |c_\btau|^{q}\ < \ \infty
$$
One can also generalize the proofs of (1) and Proposition 1 and show that 
$D(2,2)<\infty$ and $D(3,3)<\infty$, thus proving Theorem 1 for $d=4$ and 
$d=5$.  

Finally, we discuss the rate of growth on $\TT^d$ of the sum
$$
\Sigma (\rho)\ =\ \sum_{\btau\in\ZZ^d;|\btau |<\rho} |c_\btau| \tag 7
$$
Conjecture HL implies that on the four-dimensional torus there exist
quantum limits for which $\Sigma(\rho)\to\infty$ as $\rho\to\infty$;
we can prove this unconditionally for the five-dimensional torus.
We prove
\proclaim{Proposition 2}
On $\TT^d$, $\Sigma(\rho)\ll\rho^{d-3}$ for $d\geq 6$.
For $d=5$, $\Sigma(\rho)\ll\rho^{3/2+\varepsilon}$ and for $d=4$, 
$\Sigma(\rho)\ll\rho^\varepsilon$ for every $\varepsilon>0$.
\endproclaim
We expect that $\Sigma(\rho)\ll\rho^{d-4+\varepsilon}$ for any
$\varepsilon>0$.
Conjecture HL implies that for $d\geq 5$ and for any $\varepsilon>0$,
$$
\overline{\lim}_{\rho\to\infty}\;
\frac{ \Sigma(\rho) }{ \rho^{d-4-\varepsilon} }\ =\ \infty
$$
we can also prove unconditionally that for $d\geq 6$,
$\overline{\lim}_{\rho\to\infty}\Sigma(\rho)/\rho^{d-5-\varepsilon} =\infty$.

Many of the above facts are consequences of a Geometric lemma which is
of independent interest.
To motivate the lemma, note that it follows from (5) that if $\varphi$ is the
eigenfunction of the Laplacian on $\TT^d$ with the eigenvalue $\lambda$, then 
the nonzero frequencies of $|\varphi|^2$ are given by the {\it chords} 
connecting the lattice points on the sphere $S$ of radius $\sqrt{\lambda}$ in 
$\RR^d$ (centered at the origin).  Also, the size of the Fourier coefficient 
$b_\btau$ is determined by the coefficients $a_\xi ,a_\eta$ corresponding to 
such pairs $(\xi ,\eta)$ of lattice points on $S$ that the chord $\xi-\eta$ 
connecting them is equal to $\btau$.
We will call such pairs {\it $\btau$-resonances} (the terminology comes from
mechanics, where it is usually assumed that $|\btau |\ll\sqrt{\lambda}$).

Given a measure $d\nu$ on $\TT^d$ with the Fourier expansion (2),
and a sequence $\varphi_j$ of eigenfunctions of the Laplacian with
(increasing) eigenvalues $\lambda_j$, by $|\varphi_j|^2\to d\nu$ weak $*$
as $j\to\infty$ we shall mean that for every $\btau$,
$b_\btau (j)\to c_\btau$ as $j\to\infty$ (for $\btau =\bzero$, $b_\btau (j)$ is
always equal to $1$, so $c_\bzero = 1$).  Thus we are interested in
$\btau$-resonances which appear on every sphere of radius
$\sqrt{\lambda_j}$ for $j>N=N(\btau)$.

To state the Geometric lemma, we give a definition.  Let $\Delta$ be a
$d$-dimensional simplex with $d+1$ vertices all lying in $\ZZ^n$ for some $n$
(i.e. a line segment for $d=1$, a triangle for $d=2$ etc).
\proclaim{Definition}
The points $\xi_1, \xi_2, \ldots , \xi_{d+1}\in \ZZ^n$ lying on a sphere in
$\RR^n$ are called {\it $\Delta$-resonant} if they form a $d$-dimensional
simplex which can be translated into $\Delta$.
\endproclaim
This definition is just a multi-dimensional generalization of
$\btau$-resonances defined above for $d=1$.   We now state the lemma.

\proclaim{Geometric lemma}
Let $\Delta_1,  \ldots ,\Delta_k$ be a collection of $d$-dimensional
simplices with vertices in $\ZZ^{d+1}\!$, and let $S_1, \ldots ,S_n,\ldots$
be an infinite sequence of $d$-dimensional spheres in $\RR^{d+1}$
centered at the origin with increasing
radii, such that on each sphere $S_n$ there are $\Delta_j$-resonant points for
every $1\leq j\leq k$.  Then the radii of the circumscribed spheres $\Omega_j$
of $\Delta_j$-s can take (at most) two different values.  
\endproclaim
For $d=1$ this just means that the line segments $\Delta_1, \ldots ,
\Delta_k$ can have at most two different lengths (hence at most two
different circles in Theorem 1).   The lemma is proved
by first reducing the statement to counting the number of solutions
of a system of Pell equations, and then using the (known) fact that this
number is finite when the coefficients of the equations are quadratically
independent.  This fact can be proved either by applying Baker's results
on linear froms in logarithms (cf. [Bak1] and [Bak2]; also, cf. [Mi] for the
proof of a similar statement); or by applying Siegel's theorem about the
finiteness of number of integer points on elliptic curves.  The second proof,
though easier, does not produce effective bounds obtained from the first proof.

The Geometric lemma proves a certain finiteness statement about the
simplices of codimension one in $\RR^n$ whose vertices are lattice points on
spheres.  One can ask whether it is possible to prove an analogous finiteness
statements for simplices of codimension two or more.  The answer to the above
question is negative.  Namely, conjecture HL implies that there exists an
infinite sequence $\Delta_j$ of simplices with vertices in $\ZZ^n$ of
codimension two, and a sequence $S_k$ of spheres in $\RR^n$ centered at the
origin of increasing radii such that on $S_k$ there are $\Delta_j$-resonant
points for $1\leq j\leq k$ (this implies the existence of quantum limits on
$\TT^3$ which are not trigonometric polynomials); an analogous statement
for simplices of codimension three can be proved unconditionally.

We can prove analogues of the Geometric lemma for lattices in $SL(n,\RR)$;
these results will be discussed elsewhere.

The main application of the Geometric lemma is Proposition 4 given below.
It is a generalization of a result of B. Connes (cf. [Con]).
To state the result, we first define certain graphs on spheres.
\proclaim{Definition}
For a $d$-dimensional sphere $S$ in $\RR^n$ and for $\rho>0$, define
$G_\rho(S)$ to be the graph with vertices at the points of $\ZZ^n$ located on
$S$ obtained by connecting two vertices whenever the distance between
them is less than $\rho$.  Also, for $\rho>\rho_1>0$ define
$G_\rho^{\rho_1}(S)$ to be the graph with the same vertices, and with two
vertices $\xi$ and $\eta$ connected whenever $\rho_1<|\xi-\eta|<\rho$.
\endproclaim
We remark that $G_\rho^{\rho_1}(S)$ coincides with $G_\rho(S)$ for
$\rho_1=0$.

We now state Connes' result:
\proclaim{Proposition 3 (Connes)}
There exists a (positive) function $g=g_d(R)\!\asymp\! R^{2/(d+2)!}$ such that
on any $d$-dimensional sphere $S$ in $\RR^n$ of radius $R$, for any
$0<\rho\rho_1$,
there exists $N\in\NN$ such that if we
take any sphere $S=S_{j_k}$ with $j_k>N$ and consider the graph
$G=G_\rho^{\rho_1}(S)$,
then the connected components of $G$ will lie on one of
$(d-2)$-dimensional spheres $\Omega_1,\Omega_2,\ldots ,\Omega_m, m=m(j_k)$
(formed by the intersections of $S$ with the subspaces of $\RR^{d+1}$ of
codimension two).
\endproclaim
Proposition 4 thus allows us to ``reduce the dimension by two''
when proving statements about the quantum limits.

The proofs of Theorem 1,  Geometric lemma and Proposition 4, and of all the
results about the quantum limits in dimensions three and four (except for
Proposition 1) appear in [Jak].  Further details of this work will be
published elsewhere.

\subheading{Acknowledgements}
This problem arose in conversations with Professor Shni\-rel\-man
and Professor Sarnak at Princeton in the Fall of 1994.
It is a pleasure for the author to thank Professors Sarnak, Rudnick, Cordoba
and Shnirelman for helpful discussions and encouragements during this research,
and Professor Bourgain for the permission to present Theorem 2.



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